Questions involving differentials (again)

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    Differentials
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Discussion Overview

The discussion revolves around the use of differentials in calculus, particularly in the context of change of variables and the derivation of kinematic equations. Participants explore the formalities and implications of manipulating differentials in these mathematical processes.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant inquires about the "change of variables using differentials trick" mentioned in a reference, seeking clarification on its formalities.
  • Another participant questions the appropriateness of manipulating differentials when deriving kinematic equations, expressing discomfort with the method.
  • A participant suggests that the substitution rule for integration may be relevant to the change of variables discussion.
  • There is a clarification regarding the integration of differentials, noting that integral signs inherently include the variable of integration.
  • One participant explains that the relationship dv = a.dt is a logical step in both integration and differentiation, highlighting the process of taking limits.

Areas of Agreement / Disagreement

Participants express varying levels of comfort and understanding regarding the manipulation of differentials, indicating that there is no consensus on the proper approach or formalities involved.

Contextual Notes

Some participants express uncertainty about the legitimacy of treating differentials in certain ways, and there are references to specific methods that may or may not align with traditional calculus practices.

autodidude
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What is the change of variables using differentials trick K&K are referring to here?

http://books.google.com.au/books?id...f variables differentials intractable&f=false

(about halfway down the page)

Are there any formalities behind this?

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Also, when people derive the kinematic equations using calculus? I notice they rely on differentials

e.g.
http://physics.info/kinematics-calculus/

The first one, they had a=dv/dt then multiplied both sides by dt and integrated with respect to that variable...perhaps it's cause I'm still not all that comfortable with playing around with differentials like that yet but it doesn't seem 'proper' to do that. Are there alternate methods that DON'T involve treating differentials like that?

Another method that canceled the differentials is shown here at the end:

I'm not sure about that either
 
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Oh yes, just found it. It looks like the substitution rule for integration...

For the second part (kinematic equations link), when they integrate the differential, don't integral signs already come with the differential, the variable that you're integrating with respect to?
 
autodidude said:
For the second part (kinematic equations link), when they integrate the differential, don't integral signs already come with the differential, the variable that you're integrating with respect to?
Are you referring to the dv = a.dt line? That is just saying that in a small interval of time, dt, the velocity increase, dv, will be a.dt. This is the logical first step whether you're integrating or differentiating. From there, you can either divide both sides by dt, then take the limit as dt tends to zero, to get the derivative; or perform a sum of dt's over a range, then take the limit to obtain an integral.
Does that help?
 
^ Yes! Thanks a lot!
 

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